Question

question 9 of 10
the diagram below shows a rectangle inside a regular hexagon. the apothem of the hexagon is 17.32 units. to the nearest square unit, what is the area of the shaded region?
apothem length: 17.32
a. 1039 square units
b. 719 square units
c. 1359 square units
d. 373 square units

Explanation:

Step1: Find area of regular hexagon

The formula for the area of a regular polygon is \( A = \frac{1}{2} \times perimeter \times apothem \). A regular hexagon has 6 equal sides. From the diagram, the side length of the hexagon (equal to the length of the rectangle's length? Wait, no—wait, the rectangle has length 20? Wait, no, maybe the side of the hexagon? Wait, actually, in a regular hexagon, the apothem \( a \) is related to the side length \( s \) by \( a = \frac{s\sqrt{3}}{2} \). But also, the perimeter \( P \) of the hexagon is \( 6s \). So first, let's find the area of the hexagon. Wait, but do we know the side length? Wait, maybe the rectangle's length is equal to the side length? Wait, the rectangle has length 20? Wait, the diagram shows a rectangle with length 20 and width 16? Wait, the shaded region is the hexagon minus the rectangle. So first, calculate area of hexagon, then area of rectangle, subtract to get shaded area.

First, area of hexagon: \( A_{hexagon} = \frac{1}{2} \times perimeter \times apothem \). Let's find the perimeter. Wait, in a regular hexagon, the apothem \( a = \frac{s\sqrt{3}}{2} \), so \( s = \frac{2a}{\sqrt{3}} \). Given \( a = 17.32 \), so \( s = \frac{2 \times 17.32}{\sqrt{3}} \). Let's compute that: \( 2 \times 17.32 = 34.64 \), divided by \( \sqrt{3} \approx 1.732 \), so \( 34.64 / 1.732 \approx 20 \). Oh! So the side length \( s = 20 \). So perimeter \( P = 6s = 6 \times 20 = 120 \). Then area of hexagon: \( \frac{1}{2} \times 120 \times 17.32 = 60 \times 17.32 = 1039.2 \).

Step2: Find area of rectangle

The rectangle has length 20 (same as side of hexagon?) and width 16 (from diagram: the rectangle has width 16). So area of rectangle \( A_{rectangle} = length \times width = 20 \times 16 = 320 \).

Step3: Subtract to find shaded area

Shaded area \( A_{shaded} = A_{hexagon} - A_{rectangle} = 1039.2 - 320 = 719.2 \), which rounds to 719.

Wait, let's check again. Wait, hexagon area: \( \frac{1}{2} \times perimeter \times apothem \). Perimeter is 6s, s=20, so perimeter=120. Apothem=17.32. So \( 0.5 \times 120 \times 17.32 = 60 \times 17.32 = 1039.2 \). Rectangle area: 20*16=320. 1039.2 - 320 = 719.2, which is approximately 719. So the answer should be 719.

Answer:

B. 719 square units